Maths Implicit differneiation

why does y= a^x differentiate to a^xlna

i dont get the logic behind it

ln(a) is a constant

Differentiate 9x you get 9

Original post by brainmatter

why does y= a^x differentiate to a^xlna

i dont get the logic behind it

Take logs of both sides:

ln(y) = x \ ln(a)

\dfrac{1}{y} \dfrac{dy}{dx} = ln(a)

\dfrac{dy}{dx} = y \ ln(a)

\dfrac{dy}{dx} = a^x \ ln(a)

[video="dailymotion;x2ezmbr"]http://www.dailymotion.com/video/x2ezmbr_if-y-a-x-dy-dx-a-xlna-implicit-differentiation-proof_school[/video]

Original post by pleasedtobeatyou

Take logs of both sides:

ln(y) = x \ ln(a)

\dfrac{1}{y} \dfrac{dy}{dx} = ln(a)

\dfrac{dy}{dx} = y \ ln(a)

\dfrac{dy}{dx} = a^x \ ln(a)

how is this taking logs of both sides, wouldn't it be a on RHS if logs of both sides was taken?

Original post by brainmatter

how is this taking logs of both sides, wouldn't it be a on RHS if logs of both sides was taken?

Apply the log function to both sides. log(y) = log(x^a)

Bring down the exponent, using the property of logs, on the RHS to get log(y) = alog(x)

You could also write a^x = e^(xlna)

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